Local adaptive observer for linear time-varying systems with parameter-dependent state matrices

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Local adaptive observer for linear time-varying systems with parameter-dependent state matrices

Romain Postoyan, Qinghua Zhang

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Romain Postoyan, Qinghua Zhang. Local adaptive observer for linear time-varying systems with

parameter-dependent state matrices. 2018. �hal-01869339�

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Local adaptive observer for linear time-varying systems with parameter-dependent state matrices

Romain Postoyan and Qinghua Zhang

Abstract— We present an adaptive observer for linear time- varying systems whose state matrix depends on unknown parameters. We first assume that the state matrix is affine in these parameters. In this case, the proposed observer generates state and parameter estimates, which exponentially converge to the plant state and the true parameter, respectively, provided a persistence of excitation condition holds and the unknown parameters lie in a neighborhood of some known nominal values. Hence, some prior knowledge on the unknown parameters is required, but not on the state. We then modify the adaptive observer and its convergence analysis to systems whose state matrix is smooth, instead of being affine, in the unknown parameters. The convergence is approximate, and no longer exponential, in this case. An example is provided to il- lustrate the results, for which the required distance between the unknown parameters and their nominal values is investigated numerically.

I. INTRODUCTION

The objective of this work is to estimate both the state and the parameters of the system

˙

x(t) = A(t, θ)x(t) +B(t)u(t)

y(t) = C(t)x(t), (1)

where1 x(t) ∈ Rnx is the state, u(t) ∈ Rnu is the input, y(t) ∈ Rny is the output, θ ∈ Rnθ is the vector of unknown constant parameters, t ≥ 0 is the time, and nx, nu, ny, nθ ∈Z>0. In other words, we aim at designing an adaptive observer for system (1). This problem arises in many practical applications, including electrochemical batteries [13] or mechanical systems [9] to cite a couple of examples, as soon as we need to estimate on-line both the unmeasured state and unknown parameters of a plant for monitoring purposes for instance.

In the majority of works on adaptive observers, the state matrix is assumed to be fully known contrary to (1), and the unknown parameters enter in the system through an additional term of the form Φ(t, u(t), y(t))θ where Φ is known. Solutions for this case can be found in e.g., [3], [4], [20], [28]–[30]. The fact thatAdepends onθin (1) is a major difference with these references, which makes the problem difficult because of the cross-terms involving θ and x(t).

There are nevertheless results in the literature, which propose adaptive observers for systems with partially unknown state

R. Postoyan is with the Universit´e de Lorraine, CNRS, CRAN, F-54000 Nancy, Franceromain.postoyan@univ-lorraine.fr.

Q. Zhang is with Inria-IFSTTAR, Campus de Beaulieu, 35042 Rennes Cedex, Franceqinghua.zhang@inria.fr.

This report corresponds to [23] with all the proofs as well as further details.

1The notation and the terminology are defined at the end of the introduc- tion.

matrixA. In [17], several designs are given for single-input single-output linear time-invariant (LTI) systems. In [18], it is explained how to optimally fit a discrete-time LTI model to any given output signals. In [21], a switched approach is proposed for LTI systems based on linear matrix inequalities (LMI) conditions. More recently, a generic approach for the state and parameter estimation of nonlinear systems was presented in [8] using a multi-observer set-up. On the other hand, adaptive observers for systems with known state matrices with an additive term of the form Φ(x)θ(see e.g., [5], [7], [11], [12]) are relevant in this context when A is affine in θ, as a simple manipulation brings system (1) to the right form.

In this work, we propose an approach, which relies on a simple idea. We essentially assume that we know a state- observer for system (1) for some known nominal param- eter value θ0. Because θ 6= θ0, the idea is to modify the original state-observer and design an adaptation law to estimate θ by relying on similar techniques as in [30].

Indeed, by adding and subtracting A(t, θ0)x(t) in the first equation in (1), we obtainx(t) =˙ A(t, θ0)x(t) +B(t)u(t) + [A(t, θ)−A(t, θ0)]x(t). Assuming A(t, θ)is affine inθ we can write A(t, θ)x(t)−A(t, θ0)x(t) = Λ(t, x(t))(θ−θ0), whereΛ(t, x(t))is a time-varying matrix linear in its second argument; this is explained in more detailed in the sequel.

Hence,x(t) =˙ A(t, θ0)x(t) +B(t)u(t) + Λ(t, x(t))(θ−θ0).

Denoting by xˆ the state estimate of the adaptive observer to be designed and x˜ = x−xˆ the associated state esti- mation error, we derive x(t) =˙ A(t, θ0)x(t) +B(t)u(t) + Λ(t,x(t))(θˆ −θ0) + Λ(t,x(t))(θ˜ −θ0), sinceΛ is linear in its second argument. NowΛ(t,x(t))ˆ is known sincexˆis the state estimate we generate. The term Λ(t,x(t))(θˆ −θ0) is therefore of the formΦ(t)(θ−θ0)whereΦ(t)is known. We are almost back to the same class of systems as in [30] where the unknown parameter isθ−θ0instead ofθ, but this is fine since we knowθ0. The difference with [30] is that we have to deal with the extra term Λ(t,x(t))(θ˜ −θ0). It appears that when ||θ−θ0|| is small, so is Λ(t,x(t))(θ˜ −θ0)˜x(t) so that stability of the estimation error may be ensured.

The point of these developments is to show that after some simple manipulations, we can write the problem in a similar, though different form, as in [30]. Still, the design of the adaptive observer is not a straightforward application of [30]

and requires appropriate modifications as well as specific assumptions. The exponential convergence of the proposed adaptive observer is guaranteed provided a persistence of excitation condition holds and ||θ−θ0|| is small enough.

Such an excitation condition is needed in almost all works

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on adaptive observers. The fact that||θ−θ0||has to be small means that the design is local in the parameter estimate, but we are free to initialize the state estimate as we wish. It can be noted that many estimation or identification schemes are also local, see [18] for instance and all the works on extended Kalman filters, see e.g., [6], [16], [24]. Afterwards, the results are extended to the case where Ais only smooth in θ, and not necessarily affine. The adaptive observer and its analysis are modified accordingly and an approximate convergence property is ensured in this case.

Compared to [17], our results apply to multi-input multi- output linear time-varying systems, as opposed to single- input single-output LTI systems, and compared to [21], we address LTV systems and our approach relies on different type of conditions (no LMI). The closest work in spirit, although it is written in discrete-time for LTI models, is the one in [18]. A key difference with [18] is that we do not need to use projection techniques to maintain the parameter estimate in some given set, and the estimates we generate cannot be trapped in the border of the given set due to projections, but always converge to the true parameter provided, again, that ||θ−θ0|| is small enough and a persistence of excitation holds. On the other hand, the adaptive observer is less computationally demanding than the supervisory observer presented in [8], which requires a large number of observers to work, and we ensure the asymptotic convergence of the estimation errors as opposed to a practical property in [8]. Finally, compared to [5], [7], [11], [12]: (i) we do not require any particular triangular or block-diagonal structure of the system ([5], [11], [12]); (ii) we provide results for the case where the state matrix is not affine in θ in which case the results in [5], [7], [11] do not apply; (iii) we do not rely on high-gain techniques ([11], [12]), which are not known to be very sensitive to noise measurements;

(iv) the observer is of smaller dimension, which is relevant for computational reasons ([5], [11], [12]). Also, we believe that the way we approach the problem has its own interest, which can be relevant for further extensions.

A similar problem is addressed in [9]. First, exponential convergence is ensured in our case whenAis affine inθ as opposed to an ultimate boundedness property in [9]. Second, the proposed solution is computationally lighter, since [9]

requires the use of both a state-observer for the nominal parameter values and of an adaptive observer, while we only need the latter.

The rest of the paper is organized as follows. The assump- tions we make on system (1) are presented in Section II. The main result is then given in Section III and an illustrative example is provided in Section IV. Section V concludes the paper. A technical lemma is provided in the appendix.

Notation and terminology. Let R := (−∞,∞), R≥0 :=

[0,∞), R>0 := (0,∞), Z≥0 :={0,1,2, . . .}, and Z>0 :=

{1,2, . . .}. The notation(x, y)stands for [x>, y>]>, where x∈Rn,y ∈ Rm, and n, m∈Z>0. The identity matrix of sizen∈Z>0is denotedIn or simplyIwhen the dimension is clear from the context. Given two matricesQ, R∈Rn×n, we writeQ≤RifR−Qis positive semi-definite,n∈Z>0.

The symbol ? stands for symmetric blocks in matrices. A time-varying matrix M(t) ∈ Rn×m is said to be bounded if there exists c ≥0 such that ||M(t)|| ≤c for anyt ≥0, where||M(t)|| := supx∈Rm\{0}||M(t)x||

||x|| , || · ||standing for the Euclidean norm when the argument is a vector. We write N(x) = O(||x||2) for N : Rn → Rm with n, m ∈ Z>0, when there exists c ≥ 0 such that ||N(x)|| ≤ c||x||2 for anyx∈Rn. The set of piecewise continuous functions from R≥0 to Rn is denoted PC(Rn),n∈Z>0. Let p∈Rn and δ∈R>0∪ {∞}, the closed ball of radiusδ centered atpis denotedB(p, δ), withB(p, δ) =Rn whenδ=∞.

II. ASSUMPTIONS

The matrices A(t, θ), B(t) andC(t) in (1) are assumed to be continuous and bounded with respect to the timet. We make the next assumption on howA(t, θ)depends onθ.

Assumption 1: The matrix A(t, θ) is affine in θ, in the sense that there exist matricesA0(t), . . . , Anθ(t)∈Rnx×nx such that, for any t ≥0, A(t, θ) =A0(t) +Pnθ

i=1Ai(t)θi,

whereθ= (θ1, . . . , θnθ).

When Assumption 1 is not satisfied, we may redefine the vector of unknown parameters to enforce it, by eventually over-parameterizing the original system. This assumption will be relaxed in Section III-C. It is important to note that the matrices A0(t), . . . , Anθ(t) in Assumption 1 are known, as these can be directly derived from the expression ofA(t, θ). These matrices are continuous and bounded with respect to the time since so isA(t, θ).

Our design implicitly relies on the knowledge of a state observer for system (1) ifθwould be known. More precisely, we make the following assumption.

Assumption 2: There exist δ1 ∈ R>0∪ {∞} and K : R≥0 × B(θ, δ1) → Rnx×ny, which is continuous and bounded, such that for any θ0 ∈ B(θ, δ1) the origin of the system x(t) = [A(t, θ˙˜ 0)−K(t, θ0)C(t)] ˜x(t) is uniformly

globally exponentially stable.

Assumption 2 essentially means that we know a set where parameter θ lies and, if we knew θ, we would be able to design a state-observer for the corresponding system. More precisely, Assumption 2 states that there exists a neigh- borhood of parameter θ, corresponding to B(θ, δ1), such that we can design a standard (Luenberger) state observer with gain K(t, θ0) for any θ0 in this set, whose solutions uniformly, globally, and exponentially converge to solutions to (1) when θ = θ0. To see it, let x(t) =˙ˆ A(t, θ0)x(t) + B(t)u(t) + K(t, θ0)(y(t) −C(t)ˆx(t)). If θ = θ0, then the estimation error system with variable x˜ = x−xˆ is

˙˜

x(t) = [A(t, θ0)−K(t, θ0)C(t)] ˜x(t)as in Assumption 2.

A necessary and sufficient condition for Assumption 2 to hold is provided next.

Lemma 1: Under Assumption 1, Assumption 2 holds if and only if there exists a continuous and bounded mapping K : R≥0 → Rnx×ny such that the origin of x(t) =˙˜

[A(t, θ)−K(t)C(t)] ˜x(t) is uniformly globally exponen- tially stable for the (unknown) true parameter valueθ.

Proof. The necessity part immediately follows from As- sumption 2. For the sufficiency, we consider the func-

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tion W(P,x)˜ := x˜>Px˜ where P is the solution to −P˙(t) = P(t)

A(t, θ) −K(t)C(t) +

A(t, θ) − K(t)C(t)>

P(t) +I. According to Theorem 4.12 in [15], such a matrixP(·)exists onR≥0, is continuously differen- tiable, bounded, symmetric, positive definite and there exist aP, aP > 0 such that aPI ≤ P(t) ≤ aPI for any t ≥ 0. Let θ0 ∈ Rnθ and t ≥ 0. Along the solutions to x(t) = [A(t, θ˙˜ 0)−K(t)C(t)] ˜x(t), W˙ (P(t),x(t)) =˜

−||˜x(t)||2 + 2˜x(t)>P(t) [A(t, θ0)−A(t, θ)] ˜x(t). Since A(t, θ)is bounded intand affine inθaccording to Assump- tion 1, there existsδ1>0such that for anyθ0such that||θ0− θ|| < δ1, W˙ (P(t),x(t))˜ ≤ −12||˜x(t)||2. Consequently, the origin of x(t) = [A(t, θ˙˜ 0)−K(t)C(t)] ˜x(t) is uniformly globally exponentially stable according to Theorem 4.10 in [15]. Hence, Assumption 2 holds withK(t, θ0) =K(t).

Lemma 1 involvesθ, which we do not know. The follow- ing lemma provides exploitable conditions for the satisfaction of Assumption 2.

Lemma 2: Suppose the following holds.

(i) Assumption 1 holds.

(ii) There exists δ1 ∈ R≥0 ∪ {∞} such that the pair (A(t, θ0), C(t, θ0)) is uniformly completely observ- able2 for any θ0∈ B(θ, δ1).

Assumption 2 is verified with K(t, θ0) = P(t, θ0)C(t)>R−1(t) where P(t, θ0) is the solution to P˙(t, θ0) = P(t, θ0)A(t, θ0)> + A(t, θ0)P(t, θ0) − P(t, θ0)C(t)>R−1(t)C(t)P(t, θ0) +Q(t)withP(0) =P0, where P0 is any symmetric, positive definite matrix, and R(t), Q(t) are any symmetric, positive definite, continuous

and bounded matrices.

Proof. Let θ0 ∈ B(θ, δ1). Since R(t) is positive definite and the pair(A(t, θ0), C(t, θ0))is uniformly completely ob- servable, the pair(A(t, θ0), C(t, θ0)R12(t))is also uniformly completely observable. Moreover, the matrix W(t0, t) in [1] is nonsingular at t0 sinceP0 is symmetric and positive definite, it is therefore nonsingular at somet1> t0by conti- nuity. Consequently, all the conditions of Theorem 3.1 in [1]

are verified, which ensures the uniform global exponential stability of the origin ofx(t) = [A(t, θ)˙˜ −K(t)C(t)] ˜x(t).

SinceA(t, θ0)is continuous inθ0 in view of Assumption 1, so isP(t, θ0) according to Corollary 6 in Chapter 2 in [2].

Therefore, K(t, θ0) is continuous in θ0 for θ0 ∈ B(θ, δ1).

Boundedness of K(t, θ0)follows from the boundedness of P(t, θ0), which is ensured in [1]. We have proved that

Assumption 2 is satisfied.

When the matrices A and C in (1) are time-invariant, it suffices to have the pair (A(θ0), C(θ0)) observable for any θ0 with ||θ−θ0|| ≤ δ1 for some δ1 ∈ R>0 ∪ {∞}

to ensure Assumption 2. Indeed, in this case, Bass-Gura formula3 leads to a gain K(θ0), which is continuous inθ0. The same applies when L(θ0) is selected to minimize a linear quadratic cost, i.e. whenL(θ0) = P(θ0)C>R−1 and P(θ0)is the solution to the Riccati equationP(θ0)A(θ0)>+

2See [26] for a definition of uniform complete observability as well as conditions to ensure it.

3See Chapter 4 in [14] for instance.

A(θ0)P(θ0)−P(θ0)C>R−1CP(θ0)+Q. It suffices to select Q and R symmetric and positive definite to ensure the continuity ofP(θ0)inθ0 according to Proposition 1 in [10], which in turn implies the continuity ofK(θ0)inθ0.

We finally make the next boundedness assumption on system (1).

Assumption 3: There exists a hyper-rectangle X ={x= (x1, . . . , xnx) ∈ Rnx : xi ∈ [ci, ci]} where ci ≤ ci ∈ R, i∈ {1, . . . , nx}, a non-empty set of inputsMu⊂ PC(Rnu), and a non-empty set of initial conditions SX ⊂Rnx, such that any solutionxto system (1) initialized inSX with input u∈ Mu satisfiesx(t)∈ X for allt≥0.

Assumption 3 means that the solutions to (1) lie in a known compact set, which can always be embedded in a closed hyper-rectangle, whenever its input lies in set Mu. Note that there is no restriction of the “size” ofX.

III. MAIN RESULT

A. Design and analysis

Like in [25], we introduce the element-wise satura- tion function σX : Rnx → Rnx defined as σX(x) :=

1,X(x1), . . . , σnx,X(xnx))withσi,X(xi) =xi when xi∈ [ci, ci], σi,X(xi) = ci when xi ≤ ci and σi,X(xi) = ci

when xi ≥ci, for any xi ∈ R, where X, ci, ci come from Assumption 3,i∈ {1, . . . , nx}. Based on Assumptions 1-2 and inspired by [30], we propose the adaptive observer

˙ˆ

x(t) = A(t, θ0)ˆx(t) +B(t)u(t) + Λ(t, σX(ˆx(t)))¯θ(t) +

K(t, θ0) +γΥ(t)Υ>(t)C>(t)Σ(t)

×(y(t)−C(t)ˆx(t))

θ(t)˙¯ = γΥ>(t)C>(t)Σ(t)(y(t)−C(t)ˆx(t))

Υ(t)˙ = [A(t, θ0)−K(t, θ0)C(t)] Υ(t) + Λ(t, σX(ˆx(t))) θ(t)ˆ = θ(t) +¯ θ0,

(2) where θ(t)¯ ∈ Rnθ is the estimate of θ − θ0, θ(t)ˆ ∈ Rnθ is therefore the estimate of θ, Υ(t) ∈ Rnx×nθ, Λ(t, z) := [A1(t)z . . . Anθ(t)z] ∈ Rnx×nθ for z ∈ Rnx withA1(t), . . . , Anθ(t) coming from Assumption 1, Σ(t) : R→Rny×ny is any symmetric, positive definite, bounded, continuous matrix,γ∈R>0, and X ⊂Rn comes from As- sumption 3. We have two degree of freedom when designing (2):Σ(t)and γ. The former has to be designed to verify a persistence of excitation condition stated next. In practice, we typically take it diagonal. Parameterγtunes the convergence speed ofθ, the bigger¯ γ, the faster the convergence, but the more sensitive to noise.

Remark 1: The use of the saturation function σX in (2) is essential to guarantee the boundedness ofΥin the proof of the next theorem, which is needed to establish the desired convergence property. This is the main difference with [30] in terms of design. Other differences reside in the assumptions

we rely on.

The next theorem ensures that the state and the parameter estimates provided by (2) exponentially converges toxand θ, provided a persistence of condition holds and||θ−θ0||is sufficiently small. Its proof is provided in Section III-B.

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Theorem 1: Consider systems (1) and (2) and suppose the following holds.

(i) Assumptions 1-3 hold.

(ii) Let δ2 ∈ R>0 ∪ {∞}, xˆ0 ∈ Rnx, θ¯0 ∈ Rnθ, Υ0 ∈ Rnx×nθ, there exist α, T > 0, which depend onxˆ0,θ¯00, such that, for anyx(0)∈ SX and input u∈ Mu, anyθ0 such that||θ−θ0|| ≤δ2, the solution ˆ

x,θ,¯ Υ to system (2) initialized at (ˆx0,θ¯00) with

||θ0−θ|| ≤δ2, verifies for anyt≥0, αI≤

Z t+T

t

Υ>(τ)C>(τ)Σ(τ)C(τ)Υ(τ)dτ. (3) There exist δ ∈ (0,min{δ1, δ2}) and c1, c2 > 0 , which depend on X,Mu,xˆ0,θ¯00, such that, if ||θ−θ0|| ≤ δ, the solution(ˆx,θ,¯ Υ)to (2) initialized at(ˆx0,θ¯00)and any solution to system (1) withx(0)∈ SX and input u∈ Mu, verify||(x(t)−x(t),ˆ θ(t)−θ)|| ≤ˆ c1e−c2t||(x(0)−xˆ0,θ(0)ˆ −

θ)|| for anyt≥0.

Item (ii) of Theorem 1 is a persistence of excitation condition on the Υ-system, like in [30]. It actually is a uniform persistence of excitation property, see Definition 2 in [19], as that the constants α and T are independent of θ0, which is essential. Indeed, if the bound δ2, and thus δ, on the norm of θ−θ0 would depend on θ0, there will be no guarantee that we can always select θ0 sufficiently close to θ such that ||θ−θ0|| ≤δ. That is also the reason why we assume the existence of the gain K(t, θ0)for any θ0 ∈ B(θ, δ1) in Assumption 2, and not for a given θ0, otherwise δ1 in Assumption 2 may depend on θ0 and the same issue would arise. On the other hand, δ depends on the initial conditions of observer (2). It is possible to ensure thatδis uniform over these initial conditions over given sets, when (3) holds for any initial conditions in these sets.

Property (3) is difficult to verify as it involves anyθ0in a set, and any solution initialized in SX with input u∈ Mu. Even if we could verify it, that would not mean that the estimates converge to the true values, as we also need for that ||θ−θ0|| to be sufficiently small, which is something we cannot check in practice. It is important to note that this difficulty is generic to any local estimation schemes, including works on extended Kalman filters where the initial conditionsx0andxˆ0 have to be close to each other (among other conditions), see e.g., [6], [16], [24]. The only way to avoid it is when we have the property that the observation error y(t)−C(t)ˆx(t) converging to zero implies that the estimation errors x(t)−x(t)ˆ and θ−θ(t)ˆ also converging to zero, see Section III.A in [22]. In practice, we can check numerically on-line whether the inequality in (3) holds for a givenθ0 and a given solutionxto (1), in which case the non-satisfaction of (3) indicates that the obtained estimate are not valid.

The fact that we need to know a set where the unknown parameters lie is a valid assumption in various applications, like electrochemical batteries for which the parameters are accurately known when the battery is produced, but slowly vary with time. Then, the interest of the adaptive observer is to track these slow variations for monitoring purposes. If we

only have some vague knowledge aboutθ, we may overcome this issue by employing a multi-observer architecture as explained in [22].

Remark 2: Observer (2) can easily be shown to be robust in the following sense, under the conditions of Theorem 1. Consider system (1) perturbed as x(t) =˙ A(t, θ)x(t) + B(t)u(t) +wx(t), y(t) = C(t)x(t) +wz(t), where wx and wz are disturbances and noise signals respectively.

The Lyapunov analysis in Section III-B almost immediately allows to conclude that the estimation error system satisfies an input-to-state stability property [27] in this case.

provided the conditions of Theorem 1 hold.

B. Proof of Theorem 1

We introduce the estimation errors x˜ :=x−xˆ and θ˜:=

θ−θ0−θ. Let¯ x(0) ∈ Rnx, t ≥0 for which the solution to (1), (2) is defined, and θ0 ∈ B(θ,min{δ1, δ2}) where δ1, δ2 come from Assumptions 2 and item (ii) of Theorem 1, respectively. In view of (1) and (2),

˙˜

x(t) = A(t, θ)x(t) +B(t)u(t)−A(t, θ0)ˆx(t)−B(t)u(t)

−Λ(t, σX(ˆx(t)))¯θ(t)−[K(t, θ0) +γΥ(t)Υ>(t)C>(t)Σ(t)

(y(t)−C(t)ˆx(t)), (4) we add and subtractA(t, θ0)x(t)and we obtain

˙˜

x(t)=[A(t, θ0)−K(t, θ0)C(t)] ˜x(t)

+ [A(t, θ)−A(t, θ0)]x(t)−Λ(t, σX(ˆx(t)))¯θ(t)

−γΥ(t)Υ>(t)C>(t)Σ(t)(y(t)−C(t)ˆx(t)).

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In view of Assumption 1 and the definition ofΛ(·,·)given after (2),

[A(t, θ)−A(t, θ0)]x(t) = Λ(t, x(t))(θ−θ0). (6) We now restrict our attention to the case wherex(0)∈ SX withu∈ Mu. Hence, x(t)∈ SX and x(t) =σX(x(t))in view of Assumption 3 and the definition ofσ, respectively.

Thus [A(t, θ)−A(t, θ0)]x(t) = Λ(t, σX(x(t)))(θ − θ0).

Consequently, in view of (5),

˙˜

x(t)=[A(t, θ0)−K(t, θ0)C(t)] ˜x(t)

+Λ(t, σX(x(t)))(θ−θ0)−Λ(t, σX(ˆx(t)))¯θ(t)

−γΥ(t)Υ>(t)C>(t)Σ(t)(y(t)−C(t)ˆx(t)).

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We add and subtractΛ(t, σX(ˆx(t)))(θ−θ0)and we obtain, by exploiting the linearity ofΛ(·,·) in its second argument and usingθ(t) =˜ θ−θ0−θ(t),¯

˙˜

x(t)=[A(t, θ0)−K(t, θ0)C(t)] ˜x(t) +Λ(t, σX(x(t))−σX(ˆx(t)))(θ−θ0) +Λ(t, σX(ˆx(t)))˜θ(t)

−γΥ(t)Υ>(t)C>(t)Σ(t)(y(t)−C(t)ˆx(t)).

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Like in [30], we introduceη := ˜x−Υ˜θ. We deduce from

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the last equation and (2),

˙

η(t)=[A(t, θ0)−K(t, θ0)C(t)] ˜x(t) +Λ(t, σX(x(t))−σX(ˆx(t)))(θ−θ0) +Λ(t, σX(ˆx(t)))˜θ(t)

−γΥ(t)Υ>(t)C>(t)Σ(t)(y(t)−C(t)ˆx(t))

−[(A(t, θ0)−K(t, θ0)C(t)) Υ(t) + Λ(t, σX(ˆx(t)))] ˜θ(t) +Υ(t)

γΥ>(t)C>(t)Σ(t)(y(t)−C(t)ˆx(t))

=[A(t, θ0)−K(t, θ0)C(t)]η(t) +Λ(t, σX(x(t))−σX(ˆx(t)))(θ−θ0).

(9) On the other hand, in view of (2) and sincex(t) =˜ η(t) + Υ(t)˜θ(t),

θ(t)˙˜ = −γΥ>(t)C>(t)Σ(t)C(t)˜x(t)

= −γΥ>(t)C>(t)Σ(t)C(t)

Υ(t)˜θ(t) +η(t) . (10) The dynamics of the (η,θ)-system in (9), (10) is the˜ same as in (14)-(15) in [30] up to the perturbative term Λ(t, σX(x(t))−σX(ˆx(t)))(θ−θ0)in (9). Because of the lat- ter, we can no longer invoke cascade arguments to conclude about the stability of the(η,θ)-system as in [30]. We will use˜ small-gain arguments instead, and the small-gain condition will be enforced by imposing θ−θ0 to be small, which explains why the convergence property we will guarantee is local and not global (in general).

Let (ˆx,θ,¯Υ) be the solution to system (2) initialized at (ˆx0,θ¯00). We define P : R≥0×Rnθ →Rnx×nx as the solution to−P˙(t, θ0) =P(t, θ0) [A(t, θ0)−K(t, θ0)C(t)]+

[A(t, θ0)−K(t, θ0)C(t)]>P(t, θ0) + I, with P(0, θ0) = P0 and P0 is symmetric and positive definite. Since θ0 belongs to the closed ball B(θ, δ1), Assumption 2 holds and A(t, θ0)−K(t, θ0)C(t) is continuous and bounded, Lemma 1 in [19] ensures that P(t, θ0) exists for allt≥0, is continuously differentiable, bounded, symmetric, positive definite and there exist aP, aP >0, which are independent of θ0, such that aPI≤P(t, θ0) ≤aPI for anyt ≥0. We defineV : (P, η)7→η>P η. Let4t≥0, in view of (9) and the definition ofP(t, θ0)(we omit the time andθ0dependencies of the solutions), V˙(P, η) = −||η||2+ 2η>PΛ(t, σX(x)− σX(ˆx))(θ−θ0), where xˆ = x−η −Υ˜θ. The definition of Λ(·,·) and the fact that the matrices A1(t), . . . , Anθ(t) are bounded according to Assumption 1 imply that there exists a constant aΛ > 0 such that ||Λ(t, z)|| ≤ aΛ||z||

for anyz ∈Rnx. Consequently, since P ≤aPI,V˙(P, η)≤

−||η||2+2aPaΛ||η||·||σX(x)−σX(ˆx)||×||θ−θ0||. From the definition ofσ,||σX(x)−σX(ˆx)|| ≤nx||x−x||ˆ =nx||x|| ≤˜ nx||η||+nx||Υ|| · ||θ||. Thus,˜

V˙(P, η) ≤ −||η||2+ 2nxaPaΛ||η||2||θ−θ0||

+2nxaPaΛ||η|| · ||Υ|| · ||θ|| · ||θ˜ −θ0||

=−[1−2nxaPaΛ||θ−θ0||]||η||2 +2nxaPaΛ||η|| · ||Υ|| · ||θ||˜

×||θ−θ0||.

(11)

4We can take anyt0here as the solutions we consider are defined for all positive times sincex(t)∈ X and in view of (2).

We now investigate theθ-system in (10). We need the next˜ claim for that purpose.

Claim 1: There exists a continuous function ν : Rnx×nθ →[1,∞), which is independent ofθ0, such that for any continuous functionxˆ:R≥0→Rnx, the corresponding solution Υto the Υ-system in (2) initialized at Υ0 is such that||Υ(t)|| ≤ν(Υ0)for anyt≥0.

Proof of Claim 1. The result follows from Assumption 2, Lemma 1 in [19] and the fact thatΛ(t, σX(ˆx)) is bounded uniformly with respect toθ0 ∈ B(θ,min{δ1, δ2}). The fact that the image of ν is in [1,∞) can always be ensured by

adding1to any bound on ||Υ(t)||.

The matrixγΥ>(t)C>(t)Σ(t)C(t)Υ(t)is continuous and bounded since so are C(t) by assumption, Σ(t) by de- sign, and Υ(t) according to Claim 1 for the boundedness.

Therefore, since θ ∈ B(θ, δ2), item (ii) of Theorem 1 is assumed to hold, Σ(t) and C(t) are bounded and Claim 1 applies, according to Lemmas 1 and 5 in [19], the solu- tionPe(t)to−˙

Pe(t) =−γPe(t)Υ>(t)C>(t)Σ(t)C(t)Υ(t)− γΥ>(t)C>(t)Σ(t)C(t)Υ(t)>

Pe(t) +I, with P(0) =e Pe0

andPe0is symmetric and positive definite, exists for allt≥0, is continuously differentiable, bounded, symmetric, positive definite and there exist a

Pe, a

Pe >0, which are independent of θ0, such thata

PeI≤Pe(t)≤a

PeI for any t≥0. Let Ve : (P ,e θ)˜ 7→θ˜>Peθ. Let˜ t≥0, in view of (10) and the definition ofPe(t), ˙

Ve(P ,e θ) =˜ −||θ||˜ 2−2γθ˜>PeΥ>(t)C>(t)Σ(t)C(t)η.

By following similar steps as above, we derive

˙

Ve(P ,e θ)˜ ≤ −||θ||˜ 2+ 2γa

PeaC||θ|| · ||Υ|| · ||η||,˜ (12) whereaC >0 is a constant such that||C>(t)Σ(t)C(t)|| ≤ aC for all t ≥ 0, which exists since C(t) and Σ(t) are bounded.

DefineU : (P,P , η,e θ)˜ 7→V(P, η)+µVe(P ,e θ)˜ withµ >0 a constant, which will be selected below. In view of (11) and (12), for anyt≥0,

U(P,˙ P , η,e θ)˜ ≤ −[1−2nxaPaΛ||θ−θ0||]||η||2 +2nxaPaΛ||Υ|| · ||θ|| · ||η|| · ||θ˜ −θ0||

−µ||θ||˜ 2+ 2µγa

PeaC||θ|| · ||Υ|| · ||η||.˜ (13) According to Claim 1,||Υ|| ≤ν(Υ0), thus

U˙(P,P , η,e θ)˜ ≤ −[1−2nxaPaΛ||θ−θ0||]||η||2 +2nxaPaΛν(Υ0)||θ|| · ||η|| · ||θ˜ −θ0||

−µ||θ||˜ 2+ 2µγa

PeaCν(Υ0)||θ|| · ||η||˜

= −D

(||η||,||θ||),˜ M(||η||,||θ||)˜ E ,

(14) where M :=

M11 M12

? M22

with M11 := 1 − 2nxaPaΛ||θ−θ0||, M12 := −nxaPaΛν(Υ0)||θ−θ0|| − µγa

PeaCν(Υ0)andM22:=µ. MatrixMis positive definite if and only if

1−2nxaPaΛ||θ−θ0||>0 [1−2nxaPaΛ||θ−θ0||]µ

nxaPaΛν(Υ0)||θ−θ0||+µγaPeaCν(Υ0)2

>0.

(15) The first inequality above is ensured as long as||θ−θ0||is sufficiently small, in particular we assume that||θ−θ0||<

(7)

1 4nxaPaΛ

so that 1−2nxaPaΛ||θ−θ0||> 12. The second inequality also holds with ||θ−θ0|| sufficiently small, by takingµ < 1

2 γa

PeaCν(Υ0)2; note that the denominator is well-defined sinceν(Υ0)≥1in view of Claim 1. As a result, there existsδ∈(0,min{δ1, δ2}), which is independent ofθ0, such that||θ−θ0|| ≤δ implies the existence ofε >0 such that, fort≥0,

U˙(P,P , η,e θ)˜ ≤ −ε||(η,θ)||˜ 2. (16) Since min{aP, µa

Pe}||(η,θ)||˜ 2 ≤U(P,P , η,e θ)˜ ≤(aP + µaPe)||(η,θ)||˜ 2, we deduce that (η(t),θ(t))˜ exponentially converges to 0 as t → ∞. More precisely, there exist c01, c02 > 0, which depend on X,Mu,xˆ0,θ¯00, such that ||(η(t),θ(t))|| ≤˜ c01e−c02t||(η(0),θ(0)||. Moreover, since˜

||˜x(t)|| ≤ ||η(t)||+||Υ(t)|| · ||θ(t)|| ≤ ||η(t)||˜ +ν(Υ0)||θ(t)||˜ according to the definition ofηand Claim 1, we deduce that there exist c1, c2 >0, which depend on X,Mu,xˆ0,θ¯00, such that ||(˜x(t),θ(t))|| ≤˜ c1e−c2t||(˜x(0),θ(0))||, which˜ ends the proof.

C. WhenA(t, θ)is not affine in θ

We study the case where A(t, θ) is smooth in θ but not necessarily affine inθ, as required in Assumption 1. Lett≥ 0, θ ∈ Rnθ and denote A(t, θ) = [aij(t, θ)](i,j)∈{1,...,nx}2. The Taylor expansion of each aij with respect to θ gives aij(t, θ) = aij(t, θ0) + ∂aij

∂θ (t, θ0)(θ−θ0) +ρij(t, θ, θ0) where5 ρij(t, θ, θ0) =O(||θ−θ0||2). As a result,

A(t, θ) = A(t, θ0) +

nθ

X

i=1

Ai(t, θ0)(θi−θ0,i) +R(t, θ, θ0), (17) where θ = (θ1, . . . , θnθ), θ0 = (θ0,1, . . . , θ0,nθ), Ak(t, θ0) := h∂a

ij

∂θk(t, θ0)i

(i,j)∈{1,...,nx}2 for k ∈ {1, . . . , nθ},R(t, θ, θ0) := [ρij(t, θ, θ0)](i,j)∈{1,...,nx}2.

In view of (17), we modify the observer in (2) as

˙ˆ

x(t) = A(t, θ0)ˆx(t) +B(t)u(t) + Λ(t, θ0, σX(ˆx(t)))¯θ(t) +

K(t, θ0) +γΥ(t)Υ>(t)C>(t)Σ(t)

×(y(t)−C(t)ˆx(t))

θ(t) =˙¯ γΥ>(t)C>(t)Σ(t)(y(t)−C(t)ˆx(t))

Υ(t) = [A(t, θ˙ 0)−K(t, θ0)C(t)] Υ(t) + Λ(t, θ0, σX(ˆx(t))) θ(t) = ¯ˆ θ(t) +θ0,

(18) where Λ(t, θ0, z) :=

A1(t, θ0)z . . . Anθ(t, θ0)z

for z ∈ Rnx. The other matrices and parameters are selected as in (2).

Proposition 1: Consider systems (1) and (18). Suppose the following holds.

(i) Assumptions 2 and 3 hold.

(ii) Item (ii) of Theorem 1 holds along the solutions to (1) and (18).

5We can write that ρij(t, θ, θ0) = O(||θθ0||2) even though ρij

depends on the timet, sinceA(t, θ)is assumed to be bounded with respect to the time, see Section II.

There exist δ, c1, c2, c3 > 0, which depend on X,Mu,xˆ0,θ¯00 such that, if ||θ − θ0|| ≤ δ, the solution (ˆx,θ,¯Υ) to (18) initialized at (ˆx0,θ¯00) and any solution to system (1) with x(0) ∈ X and input u ∈ Mu are such that ||(x(t) − x(t),ˆ θ(t)ˆ − θ)|| ≤ c1e−c2t||(η(0),θ(0))||˜ +c3||θ−θ0||2. Sketch of proof. The proof follows the same steps as the proof of Theorem 1. Lett≥0 andθ0∈ B(θ,min{δ1, δ2}) where δ1, δ2 come from Assumptions 2 and item (ii) of Theorem 1, respectively. We first notice that instead of (6), we have [A(t, θ)−A(t, θ0)]x(t) = Pnθ

i=1Ai(t, θ0)(θi − θ0,i)x(t) + R(t, θ, θ0)x(t) = Λ(t, θ0, x(t))(θ − θ0) + R(t, θ, θ0)x(t)in view of (17) and the definition ofΛ. Thus, by following the proof of Theorem 1, we obtain the next equation instead of (9) fort≥0,x(t)∈ X withu∈ Mu,

˙

η(t) = [A(t, θ0)−K(t, θ0)C(t)]η(t)

+Λ(t, θ0, σX(x(t))−σX(ˆx(t)))(θ−θ0) +R(t, θ, θ0)x(t),

(19) and (10) still holds.

Letxbelong toSX with input u∈ Mu, and(ˆx,θ,¯Υ) be the solution to system (2) initialized at(ˆx0,θ¯00). Lett≥ 0. The same Lyapunov analysis as in the proof of Theorem 1, leads to the next equation instead of (16) when||θ−θ0|| ≤δ for someδ >0

U˙(P,P , η,e θ)˜ ≤ −ε||(η,θ)||˜ 2+ 2ηTP(t, θ0)R(t, θ, θ0)x.

(20) Since P(t, θ0) ≤ aPI, R(t, θ, θ0) = [ρij(t, θ, θ0)](i,j)∈{1,...,nx}2andρij(t, θ, θ0) =O(||θ−θ0||2), andx(t)∈ X according to Assumption 3, there existsa≥0 such that2||P(t, θ0)R(t, θ, θ0)x|| ≤a||θ−θ0||2. Hence,

U˙(P,P , η,e θ)˜ ≤ −ε||(η,θ)||˜ 2+a||η|| · ||θ−θ0||2

≤ −ε||(η,θ)||˜ 2+a||(η,θ)|| · ||θ˜ −θ0||2 (21) from which we deduce, using a||(η,θ)|| · ||θ˜ −θ0||2

ε

2||(η,θ)||˜ 2+2εa2||θ−θ0||4,

U˙(P,P , η,e θ)˜ ≤ −ε2||(η,θ)||˜ 2+2εa2||θ−θ0||4. (22) Since min{aP, µa

Pe}||(η,θ)||˜ 2 ≤ U(P,P , η,e θ)˜ ≤ (aP +µa

Pe)||(η,θ)||˜ 2, we deduce from (22) by applying the comparison lemma (see Lemma 3.4 in [15]) that

||(η(t),θ(t))|| ≤˜ c1e−c2t||(η(0),θ(0))||˜ +c3||θ−θ0||2 for some c1, c2, c3 > 0. The desired result for x˜ then follows by notingx˜ =η+ Υ˜θ and thatΥis bounded according to

Claim 1.

As already mentioned, Assumption 2 implies that we know a state observer for system (1) when θ = θ0. If we would implement this classical (non-adaptive) observer on system (1) withθ6=θ0, the state estimate would converge toxup to an error of the order of||θ−θ0||, and, trivially, the parameter estimation error would beθ−θ0. Proposition 1 shows that these properties can be improved by employing the adaptive observer (18), which provides estimates with errors of the order of||θ−θ0||2, after a sufficiently long time. We are not

(8)

Fig. 1. Coupled mass-spring system.

able to ensure the asymptotic convergence of the estimates of the true values, contrary to Section III-A, because of the perturbative termR(t, θ, θ0)in (17).

IV. ILLUSTRATIVE EXAMPLE

Consider the mass-spring system with two elements de- picted in Figure 1. The system is be modeled by (1) with: x = (x1, x2, x3, x4) ∈ R4, where x1, x3 are the displacements of the first and the second mass from their equilibrium and x2, x4 are the velocity of the first and the second mass, respectively; u is in the input applied to the second mass;y= (x1, x2)∈R2, which means that only the variables of the first mass are measured. The state matrix isA=

0 1 0 0

km1

1mb1

1 θ

m1 0

0 0 0 1

θ

m2 0 −k2m

2mb2

2

,m1= 100 and m2 = 100 are the masses, k1 = 5 and k2 = 10 are the spring stiffness as shown in Figure 1, θ >0 is the stiffness of the middle spring, which is assumed to be unknown and b1 = 0.1 andb2= 0.4 are the damping coefficients. Matrix B is (0 0 0m1

2)> and C =

1 0 0 0 0 1 0 0

. Assumption

1 holds with A0 =

0 1 0 0

mk1

1mb1

1 0 0

0 0 0 1

0 0 −mk2

2mb2

2

 and

A1 =

0 0 0 0

m1

1 0 m1

1 0

0 0 0 0

1

m2 0 −m1

2 0

. Assumption 2 is verified as(A, C)is observable as θ6= 0. Takingu(t) = 10 sin(10t) for any t ≥ 0, Assumption 3 is satisfied since the infinity norm of this input is bounded andAis Hurwitz.

We have designed the adaptive observer as in (2) with γ = 103, Σ(t) = 103 and K such that the eigenvalues of A−KC are(−1,−1.5,−2,−2.5). We have run simulations withθ= 15, and we have considered different values ofθ0, x(0),x(0),ˆ θ(0)¯ andΥ(0). Figures 2 and 3 show that the state and parameter estimates do track the state and parameter of system (1), respectively, whenθ0 = 20, x(0) = (1,0,2,0), ˆ

x(0) = (0,0,0,0), θ(0) = 0¯ and Υ0 = 0. We have then varied the latter. Simulations suggest that the convergence of the adaptive observer may be independent of the initial conditions of x, ˆx, θ¯ and Υ: only θ0 seems to matter.

The asymptotic convergence of the estimation errors is seen wheneverθ0∈[θ−11, θ+ 75]. This means thatθ0 does not need to be very close toθfor the adaptive observer to work for this example.

Fig. 2. Norm of the state estimation error.

Fig. 3. Parameterθ,θ0and the estimateθ.ˆ

V. CONCLUSION

We presented an adaptive observer for linear time-varying system whose state matrix A(t, θ) depends on unknown parameters. WhenA(t, θ)is affine inθ, the proposed scheme ensures the exponential convergence of the estimates to the true values provided some initial guess of the unknown parameter is sufficiently closed to the latter and a persis- tence of excitation holds. When A(t, θ) is only smooth in θ, a modified version of the observer has been proposed, which ensures the approximate convergence to zero of the estimation errors.

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